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The Laurent phenomenon

Sergey Fomin, Andrei Zelevinsky

TL;DR

This paper develops a unified framework for the Laurent phenomenon in birational dynamics generated by Laurent-polynomial maps. The core tool, the Caterpillar Lemma, yields Laurentness for a broad class of generalized exchange patterns, including cyclic and homogeneous constructions, by verifying a small set of coprimality and substitution conditions. Applying the framework to one-, two-, and three-dimensional recurrences (cube, octahedron, knight, Gale–Robinson, Somos) produces Laurentness and, in many cases, integrality results, addressing several longstanding conjectures. The work also situates these recurrences within a cluster-algebra perspective and conjectures nonnegative coefficient properties, suggesting wide implications for combinatorics, frieze patterns, and number-wall constructions.

Abstract

A composition of birational maps given by Laurent polynomials need not be given by Laurent polynomials; however, sometimes---quite unexpectedly---it does. We suggest a unified treatment of this phenomenon, which covers a large class of applications. In particular, we settle in the affirmative a conjecture of D$.$Gale and R$.$Robinson on integrality of generalized Somos sequences, and prove the Laurent property for several multidimensional recurrences, confirming conjectures by J$.$Propp, N$.$Elkies, and M$.$Kleber.

The Laurent phenomenon

TL;DR

This paper develops a unified framework for the Laurent phenomenon in birational dynamics generated by Laurent-polynomial maps. The core tool, the Caterpillar Lemma, yields Laurentness for a broad class of generalized exchange patterns, including cyclic and homogeneous constructions, by verifying a small set of coprimality and substitution conditions. Applying the framework to one-, two-, and three-dimensional recurrences (cube, octahedron, knight, Gale–Robinson, Somos) produces Laurentness and, in many cases, integrality results, addressing several longstanding conjectures. The work also situates these recurrences within a cluster-algebra perspective and conjectures nonnegative coefficient properties, suggesting wide implications for combinatorics, frieze patterns, and number-wall constructions.

Abstract

A composition of birational maps given by Laurent polynomials need not be given by Laurent polynomials; however, sometimes---quite unexpectedly---it does. We suggest a unified treatment of this phenomenon, which covers a large class of applications. In particular, we settle in the affirmative a conjecture of DGale and RRobinson on integrality of generalized Somos sequences, and prove the Laurent property for several multidimensional recurrences, confirming conjectures by JPropp, NElkies, and MKleber.

Paper Structure

This paper contains 5 sections, 10 theorems, 93 equations, 8 figures.

Table of Contents

  1. Introduction
  2. The Caterpillar Lemma
  3. One-dimensional recurrences
  4. Two- and three-dimensional recurrences
  5. Homogeneous exchange patterns

Key Result

Theorem 1.2

Let $H_{\rm init} = \{ (a,b,c)\in\mathcal{H}\,:\,(a-1,b-1,c-1)\notin\mathcal{H} \}$. For every $(i,j,k)\in\mathcal{H}$, the entry $y_{i,j,k}$ is a Laurent polynomial with coefficients in $\mathbb{Z}[\alpha,\beta,\gamma]$ in the initial entries $y_{a,b,c}$, for $(a,b,c) \in H_{\rm init}$.

Figures (8)

  • Figure 1: The "caterpillar" tree $\mathbb{T}_{n,m}$, for $n=4$, $m=8$
  • Figure 2: Constructing a caterpillar; $n=4$.
  • Figure 4: The initial cluster and the equivalence classes $\left\langle h\right\rangle$
  • Figure 5: Indexing set $H_9$
  • Figure 6: The cube recurrence
  • ...and 3 more figures

Theorems & Definitions (27)

  • Example 1.1
  • Theorem 1.2
  • Example 1.3
  • Theorem 1.4
  • Example 1.5
  • Theorem 1.6
  • Example 1.7
  • Theorem 1.8
  • Example 1.9
  • Theorem 1.10
  • ...and 17 more